3.137 \(\int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx\)

Optimal. Leaf size=60 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

[Out]

a*e^2*x + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (c
*f^2*x^11)/11

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Rubi [A]  time = 0.0737615, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)*(e + f*x^4)^2,x]

[Out]

a*e^2*x + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (c
*f^2*x^11)/11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} + e^{2} \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)*(f*x**4+e)**2,x)

[Out]

2*a*e*f*x**5/5 + a*f**2*x**9/9 + c*e**2*x**3/3 + 2*c*e*f*x**7/7 + c*f**2*x**11/1
1 + e**2*Integral(a, x)

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Mathematica [A]  time = 0.00401419, size = 60, normalized size = 1. \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)*(e + f*x^4)^2,x]

[Out]

a*e^2*x + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (c
*f^2*x^11)/11

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Maple [A]  time = 0.001, size = 51, normalized size = 0.9 \[ a{e}^{2}x+{\frac{c{e}^{2}{x}^{3}}{3}}+{\frac{2\,aef{x}^{5}}{5}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{a{f}^{2}{x}^{9}}{9}}+{\frac{c{f}^{2}{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)*(f*x^4+e)^2,x)

[Out]

a*e^2*x+1/3*c*e^2*x^3+2/5*a*e*f*x^5+2/7*c*e*f*x^7+1/9*a*f^2*x^9+1/11*c*f^2*x^11

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Maxima [A]  time = 1.36468, size = 68, normalized size = 1.13 \[ \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{2}{7} \, c e f x^{7} + \frac{2}{5} \, a e f x^{5} + \frac{1}{3} \, c e^{2} x^{3} + a e^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e)^2*(c*x^2 + a),x, algorithm="maxima")

[Out]

1/11*c*f^2*x^11 + 1/9*a*f^2*x^9 + 2/7*c*e*f*x^7 + 2/5*a*e*f*x^5 + 1/3*c*e^2*x^3
+ a*e^2*x

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Fricas [A]  time = 0.197068, size = 1, normalized size = 0.02 \[ \frac{1}{11} x^{11} f^{2} c + \frac{1}{9} x^{9} f^{2} a + \frac{2}{7} x^{7} f e c + \frac{2}{5} x^{5} f e a + \frac{1}{3} x^{3} e^{2} c + x e^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e)^2*(c*x^2 + a),x, algorithm="fricas")

[Out]

1/11*x^11*f^2*c + 1/9*x^9*f^2*a + 2/7*x^7*f*e*c + 2/5*x^5*f*e*a + 1/3*x^3*e^2*c
+ x*e^2*a

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Sympy [A]  time = 0.059471, size = 60, normalized size = 1. \[ a e^{2} x + \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)*(f*x**4+e)**2,x)

[Out]

a*e**2*x + 2*a*e*f*x**5/5 + a*f**2*x**9/9 + c*e**2*x**3/3 + 2*c*e*f*x**7/7 + c*f
**2*x**11/11

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GIAC/XCAS [A]  time = 0.209684, size = 68, normalized size = 1.13 \[ \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{2}{7} \, c f x^{7} e + \frac{2}{5} \, a f x^{5} e + \frac{1}{3} \, c x^{3} e^{2} + a x e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e)^2*(c*x^2 + a),x, algorithm="giac")

[Out]

1/11*c*f^2*x^11 + 1/9*a*f^2*x^9 + 2/7*c*f*x^7*e + 2/5*a*f*x^5*e + 1/3*c*x^3*e^2
+ a*x*e^2